Binary Reactive Adsorbate on a Random Catalytic Substrate

We study the equilibrium properties of a model for a binary mixture of catalytically-reactive monomers adsorbed on a two-dimensional substrate decorated by randomly placed catalytic bonds. The interacting $A$ and $B$ monomer species undergo continuous exchanges with particle reservoirs and react ($A + B \to \emptyset$) as soon as a pair of unlike particles appears on sites connected by a catalytic bond. For the case of annealed disorder in the placement of the catalytic bonds this model can be mapped onto a classical spin model with spin values $S = -1,0,+1$, with effective couplings dependent on the temperature and on the mean density $q$ of catalytic bonds. This allows us to exploit the mean-field theory developed for the latter to determine the phase diagram as a function of $q$ in the (symmetric) case in which the chemical potentials of the particle reservoirs, as well as the $A-A$ and $B-B$ interactions are equal.Comment: 12 pages, 4 figure

Helix or Coil? Fate of a Melting Heteropolymer

We determine the probability that a partially melted heteropolymer at the melting temperature will either melt completely or return to a helix state. This system is equivalent to the splitting probability for a diffusing particle on a finite interval that moves according to the Sinai model. When the initial fraction of melted polymer is f, the melting probability fluctuates between different realizations of monomer sequences on the polymer. For a fixed value of f, the melting probability distribution changes from unimodal to a bimodal as the strength of the disorder is increased.Comment: 4 pages, 5 figure

Survival probability of a particle in a sea of mobile traps: A tale of tails

We study the long-time tails of the survival probability $P(t)$ of an $A$ particle diffusing in $d$-dimensional media in the presence of a concentration $\rho$ of traps $B$ that move sub-diffusively, such that the mean square displacement of each trap grows as $t^{\gamma}$ with $0\leq \gamma \leq 1$. Starting from a continuous time random walk (CTRW) description of the motion of the particle and of the traps, we derive lower and upper bounds for $P(t)$ and show that for $\gamma \leq 2/(d+2)$ these bounds coincide asymptotically, thus determining asymptotically exact results. The asymptotic decay law in this regime is exactly that obtained for immobile traps. This means that for sufficiently subdiffusive traps, the moving $A$ particle sees the traps as essentially immobile, and Lifshitz or trapping tails remain unchanged. For $\gamma > 2/(d+2)$ and $d\leq 2$ the upper and lower bounds again coincide, leading to a decay law equal to that of a stationary particle. Thus, in this regime the moving traps see the particle as essentially immobile. For $d>2$, however, the upper and lower bounds in this $\gamma$ regime no longer coincide and the decay law for the survival probability of the $A$ particle remains ambiguous

Equilibrium Properties of A Monomer-Monomer Catalytic Reaction on A One-Dimensional Chain

We study the equilibrium properties of a lattice-gas model of an $A + B \to 0$ catalytic reaction on a one-dimensional chain in contact with a reservoir for the particles. The particles of species $A$ and $B$ are in thermal contact with their vapor phases acting as reservoirs, i.e., they may adsorb onto empty lattice sites and may desorb from the lattice. If adsorbed $A$ and $B$ particles appear at neighboring lattice sites they instantaneously react and both desorb. For this model of a catalytic reaction in the adsorption-controlled limit, we derive analytically the expression of the pressure and present exact results for the mean densities of particles and for the compressibilities of the adsorbate as function of the chemical potentials of the two species.Comment: 19 pages, 5 figures, submitted to Phys. Rev.

Exactly Solvable Model of Monomer-Monomer Reactions on a Two-Dimensional Random Catalytic Substrate

We present an \textit{exactly solvable} model of a monomer-monomer $A + B \to \emptyset$ reaction on a 2D inhomogeneous, catalytic substrate and study the equilibrium properties of the two-species adsorbate. The substrate contains randomly placed catalytic bonds of mean density $q$ which connect neighboring adsorption sites. The interacting $A$ and $B$ (monomer) species undergo continuous exchanges with corresponding adjacent gaseous reservoirs. A reaction $A + B \to \emptyset$ takes place instantaneously if $A$ and $B$ particles occupy adsorption sites connected by a catalytic bond. We find that for the case of \textit{annealed} disorder in the placement of the catalytic bonds the reaction model under study can be mapped onto the general spin $S = 1$ (GS1) model. Here we concentrate on a particular case in which the model reduces to an exactly solvable Blume-Emery-Griffiths (BEG) model (T. Horiguchi, Phys. Lett. A {\bf 113}, 425 (1986); F.Y. Wu, Phys. Lett. A, {\bf 116}, 245 (1986)) and derive an exact expression for the disorder-averaged equilibrium pressure of the two-species adsorbate. We show that at equal partial vapor pressures of the $A$ and $B$ species this system exhibits a second-order phase transition which reflects a spontaneous symmetry breaking with large fluctuations and progressive coverage of the entire substrate by either one of the species.Comment: 4 pages, 2 figures, submitted to Phys. Rev. Let

DIFFUSIVE TRANSPORT IN A ONE DIMENSIONAL DISORDERED POTENTIAL INVOLVING CORRELATIONS

This article deals with transport properties of one dimensional Brownian diffusion under the influence of a correlated quenched random force, distributed as a two-level Poisson process. We find in particular that large time scaling laws of the position of the Brownian particle are analogous to the uncorrelated case. We discuss also the probability distribution of the stationary flux going through a sample between two prescribed concentrations, which differs significantly from the uncorrelated case.Comment: 9 pages, figures are not include

Entropy, non-ergodicity and non-Gaussian behaviour in ballistic transport

Ballistic transportation introduces new challenges in the thermodynamic properties of a gas of particles. For example, violation of mixing, ergodicity and of the fluctuation-dissipation theorem may occur, since all these processes are connected. In this work, we obtain results for all ranges of diffusion, i.e., both for subdiffusion and superdiffusion, where the bath is such that it gives origin to a colored noise. In this way we obtain the skewness and the non-Gaussian factor for the probability distribution function of the dynamical variable. We put particular emphasis on ballistic diffusion, and we demonstrate that in this case, although the second law of thermodynamics is preserved, the entropy does not reach a maximum and a non-Gaussian behavior occurs. This implies the non-applicability of the central limit theorem.Comment: 9 pages, 2 figure

Random walk generated by random permutations of {1,2,3, ..., n+1}

We study properties of a non-Markovian random walk $X^{(n)}_l$, $l =0,1,2, >...,n$, evolving in discrete time $l$ on a one-dimensional lattice of integers, whose moves to the right or to the left are prescribed by the \text{rise-and-descent} sequences characterizing random permutations $\pi$ of $[n+1] = \{1,2,3, ...,n+1\}$. We determine exactly the probability of finding the end-point $X_n = X^{(n)}_n$ of the trajectory of such a permutation-generated random walk (PGRW) at site $X$, and show that in the limit $n \to \infty$ it converges to a normal distribution with a smaller, compared to the conventional P\'olya random walk, diffusion coefficient. We formulate, as well, an auxiliary stochastic process whose distribution is identic to the distribution of the intermediate points $X^{(n)}_l$, $l < n$, which enables us to obtain the probability measure of different excursions and to define the asymptotic distribution of the number of "turns" of the PGRW trajectories.Comment: text shortened, new results added, appearing in J. Phys.

Self-Consistent Model of Annihilation-Diffusion Reaction with Long-Range Interactions

We introduce coarse-grained hydrodynamic equations of motion for diffusion-annihilation system with a power-law long-range interaction. By taking into account fluctuations of the conserved order parameter - charge density - we derive an analytically solvable approximation for the nonconserved order parameter - total particle density. Asymptotic solutions are obtained for the case of random Gaussian initial conditions and for system dimensionality $d \geq 2$. Large-t, intermediate-t and small-t asymptotics were calculated and compared with existing scaling theories, exact results and simulation data.Comment: 22 pages, RevTEX, 1 PostScript figur

Universality of the Wigner time delay distribution for one-dimensional random potentials

We show that the distribution of the time delay for one-dimensional random potentials is universal in the high energy or weak disorder limit. Our analytical results are in excellent agreement with extensive numerical simulations carried out on samples whose sizes are large compared to the localisation length (localised regime). The case of small samples is also discussed (ballistic regime). We provide a physical argument which explains in a quantitative way the origin of the exponential divergence of the moments. The occurence of a log-normal tail for finite size systems is analysed. Finally, we present exact results in the low energy limit which clearly show a departure from the universal behaviour.Comment: 4 pages, 3 PostScript figure